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Connecting Fisher information to bulk entanglement in holography

Souvik Banerjee, Johanna Erdmenger, Debajyoti Sarkar

TL;DR

The paper proposes a holographic dual for the Fisher information metric of mixed boundary states, identifying it with a finite, regularized bulk volume under the Ryu-Takayanagi surface for spherical regions. It demonstrates that linear stress-tensor perturbations do not contribute at leading order, while quadratic perturbations produce finite volume corrections that scale as $R^{2d}$ (and, for scalar perturbations, as $R^{2\Delta}$), yielding a well-defined Fisher information via a second derivative of a regulated volume. A key result is the connection between leading 1/N corrections to the Fisher information and bulk entanglement entropy, with the quantum part identified with the bulk modular Hamiltonian, while the classical part maps to canonical energy. Field-theory methods via replicas and OPE corroborate the holographic findings, and the work suggests important implications for bulk reconstruction and the relationship between information metrics and gravitational dynamics, outside the semiclassical limit.

Abstract

In the context of relating AdS/CFT to quantum information theory, we propose a holographic dual of Fisher information metric for mixed states in the boundary field theory. This amounts to a holographic measure for the distance between two mixed quantum states. For a spherical subregion in the boundary we show that this is related to a particularly regularized volume enclosed by the Ryu-Takayanagi surface. We further argue that the quantum correction to the proposed Fisher information metric is related to the quantum correction to the boundary entanglement entropy. We discuss consequences of this connection.

Connecting Fisher information to bulk entanglement in holography

TL;DR

The paper proposes a holographic dual for the Fisher information metric of mixed boundary states, identifying it with a finite, regularized bulk volume under the Ryu-Takayanagi surface for spherical regions. It demonstrates that linear stress-tensor perturbations do not contribute at leading order, while quadratic perturbations produce finite volume corrections that scale as (and, for scalar perturbations, as ), yielding a well-defined Fisher information via a second derivative of a regulated volume. A key result is the connection between leading 1/N corrections to the Fisher information and bulk entanglement entropy, with the quantum part identified with the bulk modular Hamiltonian, while the classical part maps to canonical energy. Field-theory methods via replicas and OPE corroborate the holographic findings, and the work suggests important implications for bulk reconstruction and the relationship between information metrics and gravitational dynamics, outside the semiclassical limit.

Abstract

In the context of relating AdS/CFT to quantum information theory, we propose a holographic dual of Fisher information metric for mixed states in the boundary field theory. This amounts to a holographic measure for the distance between two mixed quantum states. For a spherical subregion in the boundary we show that this is related to a particularly regularized volume enclosed by the Ryu-Takayanagi surface. We further argue that the quantum correction to the proposed Fisher information metric is related to the quantum correction to the boundary entanglement entropy. We discuss consequences of this connection.

Paper Structure

This paper contains 12 sections, 76 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proposing a holographic dual for Fisher information metric
  3. Stress-tensor perturbations
  4. At linear order in the stress-energy tensor
  5. At quadratic order in the stress-energy tensor
  6. Scalar perturbations
  7. Connection to canonical energy and boundary relative entropy
  8. Justification from field theory
  9. Fisher information and bulk entanglement
  10. Bulk entanglement entropy and quantum canonical energy
  11. Canonical energy and bulk modular Hamiltonian
  12. Conclusions and outlook

Figures (1)

  • Figure 1: At the boundary CFT$_d$ of the global AdS$_{d+1}$ cylinder, we have a disc shaped region $R$ denoted by $AB$ (red line, color online). The dashed (black) line $\gamma$ represents the RT surface which divides the bulk region into two subregions $R_b$ and $R_b^c$. The area of this minimal surface gives the leading semiclassical term of the total boundary entanglement entropy $S_{EE}$. The $\mathcal{O}(G^0)$ term of bulk entanglement entropy of the region $R_b$ is a measure of the first-order quantum correction term $S_{EE,q}$ of $S_{EE}$.